The LEDs are turning on one by one, and the goal is to hit button on the 4th LED.

How are the LEDs being turned on? If there is some detectable pattern, you can determine when the switch is pressed (using millis()), and when the LED is lit (ditto), and determine the time between the two events. It is up to you to categorize the interval as too early, early, on time, late, or too late.